4. Quadratic Equations Mathematics class 10 exercise Exercise 4.4
4. Quadratic Equations Mathematics class 10 exercise Exercise 4.4 ncert book solution in english-medium
NCERT Books Subjects for class 10th Hindi Medium
Exercise 4.1
Exercise 4.1
Q1. Find the roots of the quadratic equations 3x2 - 7x + 4 = 0 by using completing square memthod
a = 3, b = -7, c = 4
Now checking for nature of roots,
D = b2 - 4ac
D = (-7)2 - 4 × 3 × 4
D = 49 - 48
D = 1
Hence D > 0
∴ There is two different and real roots
Q2.
a = 3, b = 7, c = 4
Now checking for nature of roots,
D = b2 - 4ac
D = (7)2 - 4 × 3 × 4
D = 49 - 48
D = 1
Hence D > 0
∴ There is two different and real roots
Exercise 4.2
Exercise: 4.2
Q1. Find the roots of the following quadratic equations by factorisation:
(i) x2 - 3x - 10 = 0,
Solution:
x2 - 3x - 10 = 0
⇒x2 - 5x + 2x - 10 = 0
⇒ x( x - 5 ) + 2(x - 5) = 0
⇒( x - 5 ) (x + 2) = 0
⇒( x - 5 ) = 0, (x + 2) = 0
|
x - 5 = 0 x = 5 |
x + 2 = 0 x = - 2 |
(ii) 2x2 + x - 6 = 0;
Solution:
⇒2x2 + 4x - 3x - 6 = 0
⇒ 2x( x + 2 ) - 3(x + 2) = 0
⇒( x + 2 ) (2x - 3) = 0
⇒( x + 2 ) = 0, (2x - 3) = 0
|
x + 2 = 0 x = - 2 |
2x - 3 = 0 2x = 3
|
(iii) √2x2 + 7x + 5√2 = 0;
Solution:
√2x + 2x + 5x + 5√2 = 0√2x(x + √2) + 5(x + √2) = 0
(x + √2) (√2x + 5) = 0;(x + √2) = 0, (√2x + 5) = 0
x = - √2, √2x = - 5
x = - 5/√2
x = -5√2/2
(iv) 2x2 - x + 1/8 =0
Solution:
| 2x2 - x + | 1 | = 0 |
| 8 |
Or ⇒16x2 - 8x + 1 = 0;
⇒16x2 - 4x - 4x + 1 = 0
⇒ 4x( 4x - 1 ) - 1(4x - 1) = 0
⇒( 4x - 1 ) (4x - 1) = 0
⇒( 4x - 1 ) = 0, (4x - 1) = 0
|
4x - 1 = 0 4x = 1
|
4x - 1 = 0 4x = 1
|
(v) 100x2 - 20x + 1 = 0;
Solution:
100x2 - 20x + 1 = 0⇒100x2 - 10x - 10x + 1 = 0
⇒ 10x( 10x - 1 ) - 1(10x - 1) = 0
⇒( 10x - 1 ) (10x - 1) = 0
⇒( 10x - 1 ) = 0, (10x - 1) = 0
|
10x - 1 = 0 10x = 1
|
10x - 1 = 0 10x = 1
|
Q2. Solve the problems given in Example 1.
Example1:
(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.
(ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was Rs 750. We would like to find out the number of toys produced on that day.
Q3. Find two numbers whose sum is 27 and product is 182.
Q4. Find two consecutive positive integers, sum of whose squares is 365.
Q5. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.Q6. A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs 90, find the number of articles produced and the cost of each article.
Exercise 4.3
Exercise: 4.3
Q1. Find the roots of the following quadratic equations, if they exist, by the method of completing the square:
(i) 2x2 - 7x + 3 = 0
Solution: a = 2, b = -7, c = 3Now checking for nature of roots,
D = b2 - 4ac
D = (-7)2 - 4 x 2 x 3
D = 49 - 24
D = 25
Hence D > 0
∴ There is two different and real roots
2x2 - 7x + 3 = 0
Dividing by a term 2 we get.
Putting A term and B term into a2-2ab+b2
(ii) 2x2 + x - 4 = 0;
Solution:
a = 2, b = 1, c = -4
Now checking for nature of roots,
D = b2 - 4ac
D = (1)2 - 4 x 2 x -4
D = 1 - (-32)
D = 1 + 32
D = 33
Hence D > 0
∴ There are two distinct and real roots
2x2 + x - 4 = 0
(iii) 4x2 + 4√3x + 3 = 0
Solution:
a =4, b = 4√3, c= 3
D = b2 - 4ac = (4√3)2 - 4 x 4 x 3
= 48 - 48 = 0
Here D = 0
Therefore, There are two real and equal roots.
4x2 + 4√3x + 3 = 0
⇒(2x)2 + 2 . 2x. √3 + (√3)2 = 0
⇒(2x + √3)2= 0
⇒ (2x + √3) (2x + √3) = 0
⇒ 2x + √3 = 0, 2x + √3 = 0
⇒ 2x = - √3, 2x = - √3
⇒ x = √3/2, x = √3/2
(iv) 2x2 + x + 4 = 0;
a = 2, b = 1, c = 4Now checking for nature of roots,
D = b2 - 4ac
D = (1)2 - 4 x 2 x 4
D = 1 - 32
D = -31
Hence D < 0
∴ There is no real root.
∴ Solution cannot be made of 2x2 + 1x + 4 = 0
Exercise 4.4
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